High resolution imaging of microscopic objects based on waves propagating in the far field meets known limitations due to their limited spectrum associated with their limited energy. These limitations of the spectrum apply in the time as well as in the spatial domain. These bandwidth limitations introduce naturally a band-stop or band rejection filter in the spatial frequency domain (SFD). The occupation of the SFD is therefore limited to an area or most often a disk of diameter twice as large as the bandwidth of the optical system bandwidth. This problem has been described in basic textbooks such as {Goodman, 1968}. Due to the band-stop BS described earlier: kx, kyεBS, the angular spectrum: complex amplitude as a function of the unit vector ŝ(kx/k,ky/k,√{square root over (k2−kx2−ky2)}/k) of the beam diffracted by the object. The components ŝx and ŝy are included in the unit disk in the x-y plane. The further limitations are due to instrumental considerations: the effective spectrum of the wavefield is further diminished by the configuration of the instrument collecting the emitted or scattered wave in a cone intercepting the pupil of the microscope objective (MO). Mathematically, the angular spectrum is limited by to the Numerical Aperture (NA) of the MO. It appears multiplied by a complex function called the “coherent transfer function” of the instrument (CTF), which is also the Fourier transform of the complex Amplitude Point Spread Function (APSF). The square of the amplitude of the APSF is the IPSF (Intensity Point Spread Function) or more commonly the PSF in the state of the art. The PSF is usually considered to qualify the intensity images. The autocorrelation of the CTF is the Fourier transform of the PSF and is usually denominated Optical Transfer Function: OTF and noted C(kx,ky), and is usually considered as a descriptor of the bandwidth of the optical instrument.
The extension of the significant spectrum in the spatial frequency domain depends both on the spectrum of the specimen itself and on the transfer function of the instrument or microscope, which in general constitutes the limiting factor to image resolution. Techniques have been developed to restore the spectrum of the specimen complex wavefield from the specimen wavefield intensity in the space domain: the problem consists in making a “guess” on the complex wavefield and adjusting the propagated intensity to the actual measured intensity. RMSE minimization scheme are developed to solve this task. In particular, iterative algorithms have been proposed for phase retrieval from intensity data {Fienup}. In particular, the so-called Gerchberg-Saxton {Gerchberg, 1972}, error reduction algorithms {Fienup, 1978} {Yang, 1981} have been adapted to solve the inverse problem posed by the determination of the complex wavefield in microscopy. In many situations, the problem appears however as ill posed. It is computer intensive and the applications in optical microscopy appear quite limited. Another approach is based on the measurement of field intensity on planes situated at a plurality of axial distances z (Teague, 1983). Quantitative phase imaging can be derived from the so-called “intensity transport equation” (Gureyev, 1995). The method has been applied successfully to various domains in microscopy (Nugent, 2001 #146). This technique, also designated by: transport intensity techniques: TIT, provides quantitative phase imaging, but has the drawback to be based on the computation of the derivative or gradient of the wavefield intensity, introducing thereby a high sensitivity to noise and artifacts. Similar remarks can be formulated for other quantitative phase microscopy techniques (Popescu, 2008) such as modified DIC, Hartmann-Shack wavefront analyzer or any analyzer derived from a similar principle, such as multi-level lateral shearing interferometers (Bon 2009), or in an other variant “spiral phase plate” microscopy (Maurer, 2008), which provide quantitative phase images by integration of a differential signal resulting from interferences or Fourier filtered pupil signal: artifacts and parasitic signals.
Digital Holographic Microscopy: DHM (Cuche, 1999) is based on the holographic approach, i.e. the determination of the complex field of the radiated wave from its interference (hologram) with some reference wave generated externally to or internally from the radiated wave itself. The complex field reconstructed from a hologram appears more robust and immune to artifacts and possibly, with some improvement, to noise. For that reason, it appears as a preferred embodiment of the proposed method, although the disclosed method apply to any complex field of the wave radiated by the specimen measured by any instrument or microscopy, in particular the above mentioned, non holographic approaches.
In the general context of optics and microscopy in particular, the problem of the resolution limit is posed in term of the smallest distance separating two distinguishable objects, generally point sources. It is well known that different criteria have been proposed for this purpose. Up to the recent times, the only characteristic of the field to be considered in these criteria has been the intensity of the field collected by the instrument and used to form the specimen image. In this context the degree of coherence has been shown to play an important role in the image resolution. Coherently illuminated imaging systems suffer from an inferior lateral resolution compared to its incoherent counterpart. This aspect is further intensified by a variety of post-processing methods to improve the image quality of incoherent light microscopy. Many 2D deconvolution methods can be applied to improve image quality, like deblurring, of incoherent imaging systems and 3D deconvolution techniques give rise to enhanced optical sectioning capability. Based on iterative expectation-maximization algorithm for maximum-likelihood deconvolution of incoherent images, even super-resolution has been demonstrated at the cost of computational power. All such efforts made deconvolution a common post-processing method for biological applications such as deconvolution of fluorescence microscopy images. Consequently, it is disclosed in the present patent how bring the conveniences of improved resolution to coherent microscopy too.
The high-resolution three-dimensional (3D) reconstruction of weakly scattering objects is of great interest for biomedical research. Diffraction tomography has been demonstrated to yield for 3D refractive index (RI) distributions of biological samples. For the use of such techniques in the field of virology and cancerology, a spatial resolution in the sub-200 nm domain is required. Consequently, experimental setups must shift to shorter wavelengths, higher numerical apertures (NA) and steeper illumination and/or sample rotation angles. However, the scaling of resolution to high-NA systems introduces strong diffraction and aberration sensitivity. The use of MO under non-design rotation conditions introduces additional experimental aberrations that may further degrade resolution. Unfortunately, the theory of diffraction tomography cannot correct for these conditions since it is based on direct filtering by an ideal Ewald sphere.
Therefore, we present a new approach that effectively reconstructs the object scattered field with high-NA and under non-design imaging conditions. Opposed to classical reconstruction methods like filtered back projection, we suggest an inverse filtering by a realistic coherent transfer function (CTF), namely 3D complex deconvolution.
More recently, as described previously, the capability of new phase microscopy, QPM and DHM in particular to image simultaneously amplitude and quantitative phase measurements makes it an attractive research tool in many fields, in particular biological research since it is marker free, non-invasive regards the light intensity and only camera shutter time limited. This innovation field in microscopy strongly motivates a revision of the concept of resolution limit in microscopy.
It is the main goal of the disclosed method to show how the consideration of the complex field, preferentially to the intensity of the field, can bring an improvement of the resolution limit in microscopy. This demonstration will be brought in the particular case of optical microscopy, but the validity of the method extends far beyond optical microscopy.
In the field of the so-called “superresolution”, the general idea is to utilize degrees of freedom that are deemed unnecessary. For example they can be in real space, in temporal domain, in spectral domain, or in polarization. Generally, these methods require to alter the experimental setup with additional modifications e.g. gratings or mechanically moving parts, giving rise to practical issues. By using the phase-retrieval method of Gerchber-Saxton, it has been tried to improve space and time multiplexing. The major contribution of new microscopy techniques which provide a full image of the complex field of the wave radiated from the specimen is that, contrary to intensity based microscopy techniques, it preserves fully the degrees of freedom associated with the wavefield, in particular the electromagnetic field. In particular, DHM offers the advantage of providing the amplitude A as well as the phase φ from the reconstructed complex field U. Time multiplexing methods combined with DHM methods have been demonstrated to work with low-NA systems but still owe the proof of scalability to high-NA. For ‘midrange’ systems of NA=0.42, a resolution improvement of nearly a factor 2 is possible with a synthetic aperture, however, requiring the usage of scanning devices. Other coherent light methods like structured illumination microscopy (SIM) use coherent excitation for intensity based fluorescence imaging. Despite of demonstrating sub-wavelength resolution by phase structuring, the complex detection is only partially used in excitation.
In a first embodiment, the invention teaches how the experimental observation of systematically occurring phase singularities in phase imaging of sub-Rayleigh distanced objects can be exploited to relate the locus of the phase singularities to the sub-Rayleigh distance of point sources, not resolved in usual diffraction limited microscopy.
In a second, preferred embodiment, the disclosed method teaches how the image resolution is improved by complex deconvolution. Accessing the object's scattered complex field—containing the information coded in the phase—and deconvolving it with the reconstructed complex transfer function (CTF) is at the basis of the disclosed method. It is taught how the concept of “Synthetic Coherent Transfer Function” (SCTF), based on Debye scalar or Vector model includes experimental parameters of MO and how the experimental Amplitude Point Spread Functions (APSF) are used for the SCTF determination. It is also taught how to derive APSF from the measurement of the complex field scattered by a nano-hole in a metallic film.
In a third, preferred embodiment, the disclosed method teaches a strategy to improve the efficiency of the complex deconvolution method based on a fine tuning of the Synthetic Coherent Transfer Function SCTF is disclosed, which is based on the definition of well defined criteria:                1) Criteria based on one side on the quality of the fit of the physical model for CTF to the experimental CTF measured with the instrument.        2) On the μ-posteriory evaluation of the quality of the deconvolved image which is the base of an iterative technique consisting in adjusting the SCTF parameters on the basis of criteria about the physical reality of the deconvolved image. In particular, the so-called “Phase flattening” postulates the constancy of the phase of the deconvolved phase image, outside the specimen image.        
In a fourth embodiment, the invention teaches how the limit of resolution can be extended to a limit of 216 or smaller. It is a further development of the symmetric singularity concept. Based on these ideas, the method indicates how to overcome wavelength or tilting angle limitations.
In a fifth embodiment, the invention teaches how the presented method can generalized to a tomographic approach that ultimately results in super-resolved three-dimensional RI reconstruction of biological samples.